Séminaire de Mathématique
# Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

We study two classes of *n*-generated quadratic algebras over a field *K*. The first is the class *C*_{n} of all *n*-generated PBW algebras with polynomial growth and finite global dimension. We show that a PBW algebra *A* is in *C*_{n} *iff* its Hilbert series is *H*_{A}(*z*) = 1/(1-*z*)^{n}. Furthermore each class *C*_{n} contains a unique (up to isomorphism) monomial algebra *A* = *K* ‹ *x*_{1}, … , *x*_{n }› / (*x*_{j} x_{i} | 1 ≤£ *i* < j £ *n*). The second is the class of *n*-generated *quantum binomial algebras A*, where the defining relations are nondegenerate square-free binomials *xy* - *c*_{xy} zt, with nonzero coefficients *c*_{xy}. Our main result shows that the following conditions are equivalent: (i) *A* is *a Yang-Baxter algebra*, that is the set of quadratic relations *R* defines canonically a solution of the Yang-Baxter equation. (ii) *A* is an Artin-Schelter regular PBW algebra. (iii) *A* is a PBW algebra with polynomial growth. (iv) *A* is a binomial skew polynomial ring. (v) The Koszul dual *A*! is a quantum Grassmann algebra.

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